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I'm having some trouble with a practice problem for my linear algebra class. The problem is as follows:

Consider the matrix

$A = \begin{bmatrix} 1 & 1 & 2 & 2\\2 & 2 & 5 & 5\\0 & 0 & 3 & 3\end{bmatrix}$

  1. Find a basis for $Nul(A)$ (which we shall refer to as $\beta$).
  2. Let $V = Nul(A)$.

Then we can define a linear transformation $T:V\rightarrow\mathbb{R}^3$ by $T(v) = Av$. Write down the matrix for $T$ in terms of the basis $\beta$ of $V$ and the standard basis $\epsilon = \{e_{1}, e_{2}, e_{3}\}$ of $\mathbb{R}^3$

I know that a basis of $Nul(A) = \{\begin{bmatrix} -1\\1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\0\\-1\\1\end{bmatrix}\}$, but I'm confused about how to go about writing the matrix for T, let alone the matrix for T in terms of the basis of $\beta$ of $V$ (though I'm fairly sure $T_{\beta} = T^{-1}\beta T$) or the standard basis. Could someone help explain this problem to me?

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  • $\begingroup$ It means you have to evaluate each element of the basis of $V$ And try to find a linear combination in terms of the basis in $R^3$. $\endgroup$ – Liddo Dec 8 '15 at 23:17
  • $\begingroup$ Are you sure you’ve stated the problem correctly? If $T$ really is a map from $V$ to $\mathbb R^3$ there’s not really anything to do since $T$ maps every vector in $V$ to $0$. $\endgroup$ – amd Dec 9 '15 at 4:16

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